Figuring Out New Frontiers

Math Congregation In Quest For Life`s Equations

November 28, 1985|By Richard F. Harris, San Francisco Examiner.

BERKELEY, CALIF. — Richard Karp clutches a paper cup and plate in his left hand, erases a small patch of blackboard with his right and scratches a few figures on the board to illustrate a point to Umesh Vazirani.

Behind them, several dozen other mathematicians nibble on cakes, sip coffee, juice and tea and talk about parallel processing, computational complexity and game theory. It is tea time at Berkeley`s Mathematical Sciences Research Institute.

Through the ample windows in the brand-new building, high over the University of California campus, is a view that includes the Golden Gate Bridge and downtown San Francisco.

The skyline is both inspiring and symbolic of what goes on at the three-year-old institute: Perched far above the noise and toil of everyday life, mathematicians from around the world come to ponder the big problems that, if solved, could change the very landscape of mathematics.

As the language of all science, mathematics supplies the means for phrasing and solving scientific problems. Mathematicians, as both puzzle solvers and poets, find esthetic satisfaction in an elegant mathematical phrase.

``All of us are driven . . . by the notion of what`s beautiful, although we hope to do things that are both beautiful and useful,`` said Karp, a professor of computer science at the University of California`s Berkeley campus.

``Even the computer scientists here are theoretical ones,`` said Irving Kaplansky, the institute`s director. Although a few computer terminals are available, the main tools of the trade are yellow paper tablets and blackboards, which are everywhere throughout the three-story building.

Every year since the center opened in 1982, top mathematicians from around the world have gathered with recent Ph.D.`s to take on a few big, nagging questions. The institute is an independent corporation funded by the National Science Foundation and sponsored by 14 Western universities.

More than 100 scientists will visit for up to a year--on sabbatical, vacation or special fellowship--to brainstorm over tea, ask questions during seminars or sit quietly in one of the 53 offices that overlook San Franciso Bay or nearby mountains.

This year`s participants come from as far as Japan, Brazil and Hungary to focus on one of two general topics.

In its short history, the institute already has had some spectacular success.

One group will look for ways to make complex mathematics more realistically represent the foibles of economics. The other will grapple with some problems that have defied solution for centuries to see whether it is even theoretically possible for computers to solve them.

Last year, New Zealand-born mathematician Vaughan Jones came to the institute to study a kind of math called operator algebra, which is used to describe the behavior of subatomic particles. In so doing, he discovered that his methods applied to a completely different field coincidentally under study: mathematical descriptions of knots.

This branch of mathematics is useful to chemists studying stringlike molecules such as DNA, which react according to their geometric arrangement. The mathematics of knots is also interesting in its own right.

Jones` discovery ``was the first major advance since the 1920s in knot polynomials (formulas),`` said Calvin Moore, who helped establish the institute and until last year served as its deputy director.

Mathematical parallels like that not only provide new tools for one area of endeavor, they also open up fundamental new questions. Is it a coincidence, or does the math reveal a rudimentary similarity between knots and subatomic particles?

Jones said his discovery has intrigued physicists working on a theory that the behavior of subatomic particles may be better thought of as the behavior of subatomic strings, in which twists and loops describe the nature of the basic constituents of matter.

Karp and mathematician Stephen Smale are coordinating one of the major research areas there this year: a topic called computational complexity.

This branch of mathematics isn`t as interested in solving problems as it is in learning why some solutions work so well--and why other problems defy solution.

A classic problem in this area is that of the traveling salesman who wants to visit all the cities in his territory with the least possible travel. The task is easy if he has just a few cities to visit, but with each added city, the problem becomes considerably harder. Using the most powerful computers in the world, mathematicians have solved this problem for only 319 cities.

That may be enough from the standpoint of traveling salesmen, but it is woefully inadequate when the same logic is applied to designing efficient computer boards, industrial processes or other tangled logistical tasks.